Section 5.5 Bases Other Than e and Applications

We can now differentiate and integrate natural logarithm functions and natural exponential functions comfortably. What do we do if we are faced with exponential or logarithmic functions in other bases? Let’s deal with differentiation first.

Section 5.5 Bases Other Than e and Applications We started that derivation with the natural logarithm because that is something we know how to differentiate. It is not too much of a stretch to see how the chain rule can be applied to extend this idea. Two examples are worked out below:

Section 5.5 Bases Other Than e and Applications What can we do with integration of such functions? We need to think about how we developed the integral for the natural exponential function. There is a missing, mystery natural log that should appear, but since the natural log of e is simply 1, it does not appear. The general pattern for integrating exponential functions looks like this:

Section 5.5 Bases Other Than e and Applications Let’s look at a couple of examples here;

Section 5.5 Bases Other Than e and Applications Let’s look at an application problem involving exponential functions:  A lake is stocked with 500 fish and their population increases according to the logistic curve described below (in this equation, t is measured in months). What is the limiting size of the fish population? At what rate is the population changing at ten months? After how many months is the population increasing most rapidly?